Modelling weather effects for impact analysis of ...

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Energy Policy

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Modelling weather effects for impact analysis of residential time-of-useelectricity pricing

Reid Miller, Lukasz Golab⁎, Catherine Rosenberg

University of Waterloo, 200 University Avenue West, Waterloo, Ontario, Canada N2L 3G1

A R T I C L E I N F O

Keywords:Time-of-use pricingEffect of weather on residential electricitydemandRegression models

A B S T R A C T

Analyzing the impact of pricing policies such as time-of-use (TOU) is challenging in the presence of confoundingfactors such as weather. Motivated by a lack of consensus and model selection details in prior work, we present amethodology for modelling the effect of weather on residential electricity demand. The best model is selectedaccording to explanatory power, out-of-sample prediction accuracy, goodness of fit and interpretability. We thenevaluate the effect of mandatory TOU pricing in a local distribution company in southwestern Ontario, Canada.We use a smart meter dataset of over 20,000 households which is particularly suited to our analysis: it containsdata from the summer before and after the implementation of TOU pricing in November 2011, and allcustomers transitioned from tiered rates to TOU rates at the same time. We find that during the summer rateseason, TOU pricing results in electricity conservation across all price periods. The average demand changeduring on-peak and mid-peak periods is −2.6% and −2.4% respectively. Changes during off-peak periods are notstatistically significant. These TOU pricing effects are less pronounced compared to previous studies, under-scoring the need for clear, reproducible impact analyses which include full details about the model selectionprocess.

1. Introduction

Pricing schemes intended to reduce peak electricity consumptionsuch as time-of-use (TOU) are becoming tractable as advanced meter-ing proliferates. The Ontario Energy Board established a three-tierTOU pricing scheme with three objectives: (i) to more accurately reflectthe wholesale market cost of electricity in the price consumers pay; (ii)to encourage electricity conservation across all hours of the day; and(iii) to shift electricity use from high-demand periods to lower-demandperiods (Ontario Energy Board, 2004). Properly evaluating the impactof such policies is critical for policy makers trying to reduce demand,reduce emissions and defer new generating capacity. However, isolat-ing the moderate effects of TOU pricing is challenging in the presenceof substantial confounding factors. For example, a mild or extremesummer may skew the estimated impact of TOU pricing if the effects ofweather are not adequately modelled.

We observe that there is no consensus in prior work for modellingweather effects and discussion of variable selection criteria is limited.To ensure reliable results, policy makers should insist on clear,reproducible impact analyses which include details of the explanatoryvariable selection process and justification for any variable transforma-tion used. To help produce such analyses, this paper presents a

methodology for modelling the effects of weather on residentialdemand in the context of pricing policies.

The crux of our methodology is to compare a number of aggregateelectricity demand models which have each modelled the effects ofweather differently. We use statistical measures of their explanatorypower, out-of-sample prediction accuracy, and goodness of fit to selecta model that is both well-performing and readily interpretable. Aftercareful analysis, we have chosen a multiple regression modellingstructure for its interpretability, tractability, and modularity. Toenumerate the possible models, we define three independent compo-nents: coincident weather (e.g., incorporating humidity and windchillin addition to temperature), delay or build-up of temperature thathousehold thermal controls react to (e.g., moving average of tempera-ture or cooling/heating degree-hours) and the non-linear relationshipof temperature with demand (e.g., piecewise linear and natural splinetransformations). We hypothesize that the effect of temperature onaggregate residential electricity demand is non-linear. Furthermore, wehypothesize that past temperature observations and coincident weatherobservations each provide additional explanatory value.

The second contribution of this paper is an application of theproposed methodology to evaluate the effects of Ontario's mandatoryTOU implementation according to two of its stated objectives: energy

http://dx.doi.org/10.1016/j.enpol.2017.03.015Received 6 May 2016; Received in revised form 17 January 2017; Accepted 6 March 2017

⁎ Corresponding author.E-mail address: [email protected] (L. Golab).

Energy Policy 105 (2017) 534–546

0301-4215/ © 2017 Elsevier Ltd. All rights reserved.

MARK

conservation and shifting consumption out of peak demand periods.We use a smart meter dataset of over 20,000 households in south-western Ontario, Canada that is particularly suited to our analysis. Ithas an adequate numbers of observations before and after theimplementation of TOU pricing. Furthermore, the local distributioncompany transitioned all customers from tiered rates to TOU rates at asingle point in time, meaning that there is no uncertainty introduced bya staggered TOU billing roll-out. Though the sample size and ratetransition are positive assets of the dataset, the sample time perioddoes not include adequate pre-TOU observations during the winter rateseason to assess its effectiveness. Given this limitation, we presentresults only for the summer TOU rate season and make conclusions inthat context.

2. Prior work

A literature review performed by Newsham and Bowker (2010)discusses the impacts of three types of dynamic pricing pilots: criticalpeak pricing, time-of-use, and peak time rebates. Their review includes13 TOU pilot studies conducted after 1997. They conclude that basicTOU pricing programs like Ontario's can expect to see residential on-peak demand change by −5%. An earlier TOU literature review byFaruqui and Sergici (2010) covering 12 TOU pilot studies concludedthat TOU pricing induces a −3% to −6% change in residential on-peakdemand. From 2010 onwards, there have been several impact studiesof mandatory TOU pricing. We summarize these recent studies as wellas several of the older ones in Table 1.

Our first observation is that results from opt-in experiments andpilot studies such as Hydro One (2008); Lifson and Miedema (1981);Ontario Energy Board et al. (2007) and Train and Mehrez (1994) areoften more pronounced than mandatory studies such as Faruqui et al.(2013b); Navigant Research and Newmarket-Tay Power Distribution(2010) and Navigant Research and Ontario Energy Board (2013). Oursecond observation is that most studies in our review either have apronounced demand shift from on-peak to off-peak hours or conserva-tion across all hours. Only two subsets of one study by Jessoe et al.(2013) showed the opposite effect. Finally, we observe that the tieredroll-out of TOU to high-use customers first, analyzed by Jessoe et al.

(2013), showed substantial flexibility to shift demand.Across these TOU studies, we observedmany different techniques being

used to model weather. When deciding on which modelling techniques toconsider in our methodology, we broadened our literature review toresidential electricity demand analysis in general. Table 2 summarizes thisbroadened literature review, grouping prior work by the technique used totransform temperature observations. An explanatory variable transforma-tion is a mathematical process that creates derived values from observedvalues. For example, a series of dry-bulb temperature observations may betransformed using humidity and wind chill to become a series of perceivedtemperatures. The derived variable would be used as input to the modellingprocedure in place of the observed variable.

Table 1Results from prior TOU electricity pricing studies.

Study Pilot Mand. Season Total Change(%)

On-Peak (%) Mid-Peak(%)

Off-Peak (%) Weekend

Hydro One (2008) Yes No summer −3.30 −3.70 NR NR NRLifson and Miedema (1981) Yes No summer −3.17 −8.84 −3.95 +2.86 NAOntario Energy Board et al. (2007) Yes No summer −6.00 −2.40 (NS) NR NR NRTrain and Mehrez (1994) Yes No full year NR −9.02 NA +6.51 NAJessoe et al. (2013) No Yes Summer −3.14a −6.09a NA −2.00a NA

Summer +0.39b +1.16b NA +0.06b NAsummer +2.64c +3.11c NA +2.4c NA

Faruqui et al. (2013b) No Yes Summer 0 to −0.45d −2.60 to−5.70

Decrease Increas NR

Winter 0 to−0.45d −1.60 to−3.20

Decrease Increase

Navigant Research and Newmarket-Tay PowerDistribution (2010)

No Yes Full year −0.66 (NS) −2.80 −1.39 +0.16 (NS) +2.21

Navigant Research and Ontario Energy Board(2013)

No Yes Summer 0 to −0.10 −3.30 −2.20 +1.20 +1.90Summershoulder

NR −2.20 −1.50 +1.50 +1.40

Winter NR −3.40 −3.90 −2.50 −1.20Winter shoulder NR −2.10 −2.30 −1.10 +0.50 (NS)

Maggiore et al. (2013) No Yes Jan–Jun NR −0.83 NA NR NAMei and Qiulan (2011) No Yes Feb–Dec increase increase NA increase NA

NR – not reported, NA – not applicable, NS – not statistically significant.a High-use customers only.b Medium-use customers only.c Low-use customers only.d Annual.

Table 2Categories of temperature transformations found in prior work, used when modellingresidential electricity demand.

Coincident weather transformations

Humidity Mountain and Lawsom (1992)Humidex Faruqui et al. (2013b)Temperature

Humidity IndexFaruqui et al. (2013a); Navigant Research and OntarioEnergy Board (2013)

Wind Speed Friedrich et al. (2014); Mountain and Lawsom (1992)Temporal transformationsLagged Observations Harvey and Koopman (1993)Heating and Cooling

Degree-DaysPardo et al. (2002); Cancelo et al. (2008)

Heating and CoolingDegree-Hours

Navigant Research and Newmarket-Tay PowerDistribution (2010)

Moving Average Mountain and Lawsom (1992)Weighted Moving

AverageFriedrich et al. (2014); Bruhns et al. (2005)

Non-linear transformationsSwitching Regression Moral-Carcedo and Vicéns-Otero (2005); Faruqui et al.

(2013b); Navigant Research and Newmarket-TayPower Distribution (2010); Navigant Research andOntario Energy Board (2013); Lifson andMiedema (1981); Train and Mehrez (1994)

Linear Regions withSmoothedTransitions

Bruhns et al. (2005); Friedrich et al. (2014); Moral-Carcedo and Vicéns-Otero (2005)

Regression Splines Engle et al. (1986); Harvey and Koopman (1993)

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We define coincident weather to be measurable weather phenom-ena which coincide with temperature observations. For example, thehumidity observed at time i is coincident with dry-bulb temperatureobserved at time i. Several studies transform temperature by takinghumidity into account via the temperature humidity index (Faruquiet al., 2013a; Navigant Research and Ontario Energy Board, 2013), theCanadian Humidex (Faruqui et al., 2013b), or by incorporatinghumidity into some other transformation of temperature (Mountainand Lawsom, 1992). Humidity may have a direct effect on load viadehumidification equipment, or an indirect effect via human percep-tion and comfort levels. Wind speed has also been incorporated intotemperature transformations (Friedrich et al., 2014; Mountain andLawsom, 1992). Wind may reduce electricity demand if customerschoose to cool their home by leaving windows open during transitionseasons. Wind chill may also affect perception of winter outdoortemperatures, inclining a customer to stay indoors.

Temporal transformations account for the delay between when anoutdoor temperature occurs to when its effects are felt within acustomer's home. Heating degree-days and cooling degree-days arederived values used to measure the prolonged heating and coolingrequirements of a home over time. They have been extended to heatingdegree-hours and cooling degree-hours, derived by summing thedifference between recent observations and a selected temperaturebreak point. For modelling long-term and mid-term analysis horizons,heating and cooling degree-days are sufficient (Pardo et al., 2002;Cancelo et al., 2008). Heating and cooling degree-hours are bettersuited to the analysis of short-term and mid-term horizons (NavigantResearch and Newmarket-Tay Power Distribution, 2010). Harvey andKoopman (1993) considered lagged hours of temperature observationsin early models of their study. Mountain and Lawsom (1992) used afour-hour moving average of recent temperatures as a component ofthe space heating index used in their model. Friedrich et al. (2014)refined work by Bruhns et al. (2005) to account for thermal transferinertia. The authors define an exponentially weighted moving averagefilter to be the smoothed temperature. Moral-Carcedo and Vicéns-Otero (2005) describe a single temperature break point as a switchingregression to model temperature's non-linear relationship with elec-tricity demand. The coefficient found for temperatures below the breakpoint represents household heating effects. The coefficient for tem-peratures above the break point represents cooling effects. It is used byFaruqui et al. (2013b); Navigant Research and Newmarket-Tay PowerDistribution (2010), and Navigant Research and Ontario Energy Board(2013). Lifson and Miedema (1981), and Train and Mehrez (1994) alsouse switching regression, but the lower region has a slope of zerobecause households in their regions of study have no heating effects.

Intuitively, the boundary between heating and cooling effects is notan abrupt break. When subjected to moderate temperatures, occupantsmay not heat or cool their home. Each household will heat or cool theirhome at different temperatures, resulting in a smoothed transitionregion when data is analyzed in aggregate (Bruhns et al., 2005;Friedrich et al., 2014; Moral-Carcedo and Vicéns-Otero, 2005).Cancelo et al. (2008) note that extreme low temperatures and extremehigh temperatures exhibit saturation of heating and cooling effects. Atthese temperatures, all household thermal controls such as spaceheaters, electric baseboard heating, fans, or air conditioning areworking constantly.

Regression splines, a widely-used explanatory variable transforma-tion in econometric literature, are capable of modelling the smoothtransitions between heating effects, mid-temperatures, cooling effects,and saturation plateaus at temperature extremes (Engle et al., 1986;Harvey and Koopman, 1993). The regression spline transformationfirst divides the range of temperatures into a number of regions. Withineach region, a polynomial function is fit to the data and constraints maybe placed on the polynomial functions to connect them at the regionboundaries.

3. Data description

List of symbolsN the number of hours in the sample periodJ the number of residential smart meters (i.e., households) in the

sampleτ the N × 1 vector of hourly, dry-bulb temperature observationsτ′ the intermediate N × 1 vector resulting from the transformation

of τ incorporating coincident weather observationsτ″ the intermediate N × 1 vector resulting from the transformation

of τ′ incorporating past observations. It representstemperature's effects over time

ϒ the N J× matrix of hourly electricity demand per householdY the N × 1 vector of hourly, aggregate residential electricity

demandX the temporal explanatory variable transformation matrixV the price explanatory variable transformation matrixT the weather explanatory variable transformation matrix

Y the N × 1 vector representing the model's estimate of Y

β0the estimated intercept term from which all other coefficientsare offset

β the vector of coefficient estimates for temporal explanatoryvariables in X

ω the vector of coefficient estimates for price explanatory variablesin V

θ the vector of coefficient estimates for weather explanatoryvariables in T

The smart meter dataset used in this paper was provided by a localdistribution company in southwestern Ontario. The observations occurover a period of 20 months, from March 1, 2011 through October 17,2012. The switch from a seasonal, flat pricing scheme to TOU pricingoccurred on November 1, 2011. The TOU rates, illustrated in Fig. 1, arecomprised of three price levels: off-peak, mid-peak and on-peak.Summer off-peak hours are 7:00 pm through 6:59 am (overnight) at6.5 ¢/kWh. Mid-peak hours are 7:00 am through 10:59 am and5:00 pm through 6:59 pm at 10 ¢/kWh. On-peak hours are 11:00 amthrough 4:59 pm at 11.7 ¢/kWh. All hours of weekends and holidaysare off-peak rates.

The data contains hourly smart meter readings from 23,670residential customers across a four-city service region. We removed3100 m with customer account changes (e.g., tenant changes).Additionally, we performed a data cleaning step to remove extremeoutliers. The maximum short-term overloading of a distributiontransformer is 300% of its nameplate rating (IEEE StandardsAssociation, 2012, Section 8.2.2). Using this as a guideline, 14 smartmeters had an hourly reading that violated the maximum short-termoverloading capacity of the transformer they were connected to andhence, were removed from the sample.

The remaining sample contains J=20,556 smart meter time series forstudy, each with up to N=14,328 data points (the number of hours in oursample period). Individual meter readings were stored with <1 Whprecision. Missing observations were stored as zero values, and nearlyall meters have at least a few missing observations over the course of thesample period. 0.46% of the individual readings were missing. Often, ameter's missing values occur as irregularly positioned gaps lasting multi-ple hours, such that data interpolation is not suitable. We consider thedata in aggregate by deriving the average household demand in each hourfrom all households. Let the variable ϒ1 represent the N J× matrix ofhousehold smart meter readings fromMarch 1, 2011 through October 17,2012. I ϒ( > 0)i j, is an indicator function that returns 1 if there exists areading during hour i for meter j. As Eq. (1) is evaluated from i=1,…,N, anN × 1 vector Y representing the aggregate electricity demand for eachhour of the sample period will be created.

1 We use regular font for scalar variables, and bold font for vector and matrix variables.

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Ii Ny

ϒ

ϒ=

∑

∑ ( > 0), = 1,…,i

jJ

i j

jJ

i j

=1 ,

=1 , (1)

The aggregate electricity demand observations fall in the range0.49 kWh–3.54 kWh and are approximately lognormally distributedwith mean 1.18 kWh and median 1.03 kWh. We use the vector Y as theresponse variable for the remainder of this study, plotted over time inFig. 2. Notice the summer air conditioning demands during thesummer months and less noticeable heating effects during winter.

We also obtained the corresponding hourly weather data from twonearby Environment Canada (2015a) monitoring stations. Weatherobservations were paired with each meter by selecting the nearestmonitoring station, all within 5–25 kilometres. Hourly observationsrecorded are dry-bulb temperature, relative humidity, dew point, winddirection, wind speed, visibility, atmospheric pressure, humidex, windchill and a weather condition description. We define τ to be an N × 1vector of hourly temperature observations averaged from the twoweather stations, weighted by the number of meters reporting nearthat station each hour. The two summers are not drastically differentfrom one another, as shown by key summary statistics in Table 3.Summer 2012 had a slightly higher median drybulb temperature of20.2 °C compared to 19.1 °C in 2011.

Throughout Section 4.3, τ will be used as input to temperaturetransformation functions which create a matrix T of dimensionN P× weather , where Pweather is the number of columns in T, deter-mined by the variable transformation applied. Left untransformed,

τT = .

4. Methodology for modelling the effects of weather

We use a multiple regression model shown in Eq. (2) to representelectricity consumption as a function of time, price and weather relatedvariables. Let Y be an N × 1 vector representing the model's estimate ofY. Let β0 be the estimated intercept term. We store the explanatoryvariables using three matrices X, V and T which represent time, priceand temperature transformations respectively. The effects of theseexplanatory variables are represented by the coefficient estimatevectors β , ω and θ fit using ordinary least squares.

β ω θβY X V T= + + +0 (2)

Our treatment of time and price explanatory variables, selectingcategorical variables for inclusion using forward selection and analysisof variance (ANOVA), follows well-established statistical learningmethods James et al. (2013), Ch.6. ANOVA performs a hypothesis testcomparing two models. The null hypothesis is that the less-complexmodel with fewer explanatory variables is sufficient to describe theresponse. The alternate hypothesis is that a more complex model isrequired. ANOVA tests whether the variance explained by an addedexplanatory variable or interaction is significantly different from theoriginal model. We direct the reader to Faraway (2002), Ch. 10 for fulldetails of the formulation and use of ANOVA. During forward selection,we begin with the null model, which contains the intercept β0 but noexplanatory variables in X, V, or T. We then fit a number of alternatemodels, each with a single explanatory variable added to X or V. Theexplanatory variable which results in an alternate model with the

Fig. 1. This chart from the Ontario Energy Board (2012) shows the 24-h schedule in a clock-like format. Off-peak prices are shown in green, mid-peak in yellow, and on-peak in orange.The summer schedule is on the left, weekend schedule in the middle (both seasons), and winter schedule on the right. (For interpretation of the references to color in this figure legend,the reader is referred to the web version of this article.)

Fig. 2. Aggregate residential electricity demand plotted as a function of time.Transparency has been used to give a sense of density.

Table 3Summary statistics for τ , the weighted average of drybulb temperatures (°C) within theservice region during summer 2011 and summer 2012 rate seasons.

Year Minimum 1st Quartile Median Mean 3rd Quartile Maximum

Summer2011

0.4 14.0 19.1 18.8 23.8 37.2

Summer2012

1.3 15.6 20.2 19.8 24.3 37.7

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lowest residual sum of squares is added to the null model. The processof adding explanatory variables one at a time is continued until theANOVA stopping condition is met. Some of the variables that wereconsidered during forward selection but were ultimately ruled out are:weather description, visibility, day-of-week, and a schoolyear indicator.

The remainder of this section presents the details of our methodol-ogy. The first step is to select time-dependent variables (Section 4.1).The second step is to select price-related variables (Section 4.2). Thethird step is to select and justify a model for weather effects, whichcomprises the bulk of our effort and contribution (Sections 4.3 and 4.4).

4.1. Time-related variables

We model hour-of-day as a categorical variable with 24 terms,represented in X by 23 sparse columns with indicators for each hour.Fig. 3 shows a box plot of electricity demand grouped by hour-of-dayand exhibits the expected patterns of user activity within the home.People use less electricity in the middle of the night from 01:00–06:00and are most active in the evening from 18:00–22:00.

Residential electricity demand also differs by day-of-week and onholidays. We are able to achieve a high level of explanatory power usingonly one degree of freedom by defining a working day indicator,similar to Møller Andersen et al. (2013); Moral-Carcedo and Vicéns-Otero (2005), such that weekends and holidays are non-working days.

The use of a working day indicator allows for meaningful variableinteractions to be fit. The main effects of each explanatory variablerepresent deviation from the sample mean and the two-way interac-tions represent deviation from their main effects. An interactionbetween two categorical factors such as hour-of-day and working dayis a sparse matrix with indicators for each unique combination of twovariables not represented by their main effects. For example, thebaseline for hour-of-day is 00:00 and the baseline for working day isworking_day=FALSE. If observation i occurs at 00:00 on a non-working day, neither variable's main effects will be added to β0. Ifobservation i is 07:00 on a non-working day, only the coefficientestimate for 07:00 will be added to β0 (i.e., main effects). If observationi is 07:00 on a working day, an interaction effect (denoted by07:00×working_day=TRUE) is added to β0 representing the deviationfrom the main effects of each variable. An example of working day×-hour-of-day interaction coefficient estimates is given in Table 4.Aggregate electricity demand begins earlier on working days, indicatedby a positive coefficient estimate that is of noticeable effect size and hasa statistically significant p-value. This is likely caused by residentialcustomers preparing for work around 07:00 or 08:00 on working days.10:00 through 17:00 on working days has a negative coefficient

estimate, likely because many residential customers are away at work.As suggested by Fig. 2, there are clear seasonal patterns during

summer and winter months. Fitting a model with a categoricalexplanatory variable for month is statistically significant and increasesAdjusted R2. However, our goal is to evaluate temperature transforma-tions used to generate T. Any explanatory variable that is collinear withthe temperature transformation matrix masks its effects, meaning thatthe estimated effects of two explanatory variables increase and decreasetogether. We check for collinearity using variance inflation factor(VIF) (Fox and Weisberg, 2011). Table 5 shows VIF values when acategorical variable for month is considered; a VIF > 5 indicatescollinearity (James et al., 2013). Using this measure, we determinethat addition of month masks the effects of temperature. For thisreason we do not include month as a categorical variable.

As a result of forward selection and the justification processdescribed above, we arrive at a desired set of temporal explanatoryvariables in X (we define the notation x p•, to represent the pth columnand all rows of X; this same notation will be used with other matricesgoing forward).

x p•, = 1 through x p•, = 23 are hour-of-day indicators representing 01-:00 through 23:00.

x p•, = 24 is a working day indicator.x p•, = 25 through x p•, = 48 are indicators representing the hour-of-da-

y×working day interaction.

Fig. 3. Plot of aggregate electricity demand grouped by hour. Note that this plot containsdata from both working and non-working days.

Table 4Coefficient estimates and p-values illustrating the intuition behind the hour-of-day×working day interaction. Starred p-values are statistically significant. The coefficientestimates will change slightly with each temperature transformation compared, but thesign, intuition and statistical significance remain applicable.

Interaction term Coefficient estimate p-value

01:00×working_day=TRUE 0.001 0.977202:00×working_day=TRUE −0.009 0.864203:00×working_day=TRUE 0.007 0.881404:00×working_day=TRUE 0.017 0.727805:00×working_day=TRUE 0.031 0.537706:00×working_day=TRUE 0.066 0.185007:00×working_day=TRUE 0.135 0.0066**

08:00×working_day=TRUE 0.152 0.0022**

09:00×working_day=TRUE −0.010 0.833610:00×working_day=TRUE −0.140 0.0048**

11:00×working_day=TRUE −0.208 0.0000***

12:00×working_day=TRUE −0.246 0.0000***

13:00×working_day=TRUE −0.261 0.0000***

14:00×working_day=TRUE −0.268 0.0000***

15:00×working_day=TRUE −0.252 0.0000***

16:00×working_day=TRUE −0.214 0.0000***

17:00×working_day=TRUE −0.153 0.0020**

18:00×working_day=TRUE −0.086 0.0817.19:00×working_day=TRUE −0.061 0.220520:00×working_day=TRUE −0.025 0.608821:00×working_day=TRUE 0.016 0.750122:00×working_day=TRUE 0.043 0.391923:00×working_day=TRUE 0.038 0.4386

Table 5VIF of explanatory variable main effects with a categorical variable for month. Note: Anatural cubic spline transformation of temperature has been used in this example,though similar results are achieved with nearly all temperature transforms discussed inSection 4.3.

Explanatory VIF Degrees of freedom

Natural Cubic Splines T 8.52 4Month 7.60 11Hour-of-Day 1.29 23Working Day 1.01 1

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4.2. Price-related variables

We use two pricing categorical variables. The first is a TOU billingindicator. The second is a categorical variable representing the localdistribution company's billing seasons: summer or winter. We wereconcerned that similar to the month categorical variable, the utility rateseasons might also be collinear with temperature. However, Table 6 showsthat the addition of pricing variables are not collinear with a temperaturetransformation as we iterate through transformations of T.

We will later saturate the price explanatory variable matrix V withinteractions in our TOU case study in Section 6 after finding a suitabletemperature transformation. To summarize, the explanatory variablesincluded in V are:

v p•, = 1 is a utility rate season indicator representing summer andwinter rates.

v p•, = 2 is a TOU active indicator representing whether customers arebilled according to flat rates or TOU rates.

4.3. Weather-related variables and transformations

To select a weather effects model, we define three steps oftemperature transformations which are used in conjunction with oneanother to generate variations of T.

1. Coincident Weather Transformations: dry-bulb temperatureor feels like temperature

2. Temporal Transformations: current observation, lagged obser-vations or moving average

3. Non-Linear Transformations: switching regression or naturalcubic splines

Our methodology iterates over all combinations of temperaturetransformations listed above. Each iteration uses a different combina-tion of transformation functions to generate the temperature transformmatrix T while holding the matrices X and V fixed. We begin eachtemperature transformation with an N × 1 vector of outdoor, dry-bulbtemperature observations τ . Algorithm 1 shows the general process fortransforming τ into T.

Algorithm 1. Overview of how temperature transformations arecombined to generate the matrix T.

1. Transform dry-bulb temperature observations τ into the vector τ′using coincident weather observations.

2. Transform the vector τ′ into the vector τ″ using a transformationwhich incorporates past observations. This transformation repre-sents temperature's effects over time.

3. Finally, use the vector τ″ as input into a transformation whichmodels the non-linear relationship between τ″ and aggregateelectricity demand Y. The result of this third step is the matrix T

used in the multiple regression model.In Section 4.3.4, we add two complex transformations to our

comparison which violate Algorithm 1: the heating/cooling degree-hour transformation and the exposure-lag-response transformation.Both transformations combine algorithm steps two and three, trans-forming τ′ directly to the matrix T.

4.3.1. Coincident weather transformationsDuring the first step of Algorithm 1, relative humidity and wind

speed are used to transform dry-bulb temperature observations to afeels like temperature comprised of heat index and wind chill valueswhere applicable. Algorithm 2 defines the feels like transformation.

Algorithm 2. The feels like temperature transformation. Formulationof heat index is described by Rothfusz (1990) and wind chillformulation is described by Environment Canada (2015b).

if τ Relative Humidityand> 27 > 40%i i then

τ Heat Index′ =i i

else if τ Wind Speedand≤ 10 > 4.8 kphi i

τ Wind Chill′ =i i

elseτ τ′ =i i

end if

If τ is left untransformed during this step, then τ′ would remain avector of dry-bulb temperature observations such that τ τ′ = .

4.3.2. Delayed effects of temperatureWe also need to account for the delay between when an outdoor

temperature occurs to when its effects are felt within a customer'shome. To assess the importance of past temperature in predictingpresent electricity consumption, Table 7 shows the correlation coeffi-cient of 0–12 lags of dry-bulb temperature τ with yi. The correlation ofyi with past temperatures suggests that there may be an underlyingtemporal process interacting with temperature.

The lagged observation transformation shown in Eq. (3) considersthe possibility that temperature's effects on electricity demand may bedelayed by a number of hours ℓ, also known as lags. The cause for thisdelay may be the time it takes an outdoor temperature to pass througha building's insulation. After the time delay, the household's thermalcontrols react. This interim transformation vector has ℓ fewer rows thanτ′ used as input, requiring that rows i = 1,…,ℓ must also be removedfrom Y, X and V.

τ τ i N″ = ′ , = (1 + ℓ),…,i i−ℓ (3)

A moving average of recent temperatures shown in Eq. (4) reflectsthe possibility that household thermal control systems are not reacting

Table 6VIF of explanatory variable main effects with addition of utility rate season and a TOUbilling indicator. Note: A natural cubic spline transformation of temperature has beenused in this example, though similar results are achieved with nearly all temperaturetransforms discussed in Section 4.3.

Explanatory VIF Degrees of freedom

Natural Cubic Splines T 3.08 4Rate Season 2.85 1TOU Active 1.08 1Hour-of-Day 1.17 23Working Day 1.00 1

Table 7Up to 5 lags of dry-bulb temperature are correlated with aggregate electricity demand atlevels comparable to dry-bulb temperature at time i.

Lagged Dry-Bulb temperature Correlation with yi

τi 0.539τi−1 0.551τi−2 0.558τi−3 0.558τi−4 0.550τi−5 0.533τi−6 0.509τi−7 0.477τi−8 0.440τi−9 0.400τi−10 0.361τi−11 0.328τi−12 0.302

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only to temperature at time i or some past time i − ℓ. Instead, thermalcontrol systems may be reacting to a number of recently experiencedtemperatures. The variable L represents the number of recent tem-peratures used in the moving average. The moving average transfor-mation vector has L − 1 fewer rows than τ′ used as input, requiring thatrows i = 1,…,ℓ must also be removed from Y, X and V.

ττ

Li L N″ =

∑ ′, = ,…,i

Liℓ=0

−1−ℓ

(4)

If τ′ is left untransformed during the second step of the temperaturetransformation Algorithm 1, then the output of the past weatherobservation transformation would be current observations τ′ such thatτ τ″ = ′.

4.3.3. Non-linear temperature effectsThe top graph of Fig. 4 shows the coefficient estimate θ fit for an

untransformed vector of dry-bulb temperature observations. It is clearthat temperature's relationship with aggregate electricity demand inour dataset is non-linear. The non-linear relationship between tem-perature and aggregate electricity demand can be approximated by anumber of linear regions. This approach is generally referred to as apiecewise linear transformation or linear splines (James et al., 2013).Moral-Carcedo and Vicéns-Otero (2005) describe piecewise linearmodels with two linear regions as switching regression, giving mean-ing for electricity demand analyses; the break point represents theswitch from heating effects to cooling effects. Eq. (5) shows thetransformation of τ″ into a column of T representing heating effects.

τξ i Nt = ( − ″ ) , = 1,…,i break i,1 + (5)

Similarly, Eq. (6) shows the transformation of τ″ into a column of Trepresenting cooling effects.

τ ξ i Nt = ( ″ − ) , = 1,…,i i break,2 + (6)

Let x max x( ) ≔ (0, )+ and let ξbreak be a temperature break pointestimated empirically (Muggeo, 2003, 2008). The fitted regression linefor this switching regression transformation is shown in the middlegraph of Fig. 4.

Moving beyond piecewise transformations, regression splines havebeen used to first divide the range of temperatures into a number ofregions. Within each region, a polynomial function is fit to the data andconstraints are placed on the polynomial functions to connect them atthe region boundaries, known as knots. Similar to switching regression,the goal of piecewise polynomial transformation is to break τ″ intoregions using break points called knots, represented by the K × 1 vectorξ. Let K be the number of knots, such that there are K + 1 regions. Foreach region, a polynomial function is used to transform observations inτ″. The bottom graph of Fig. 4 illustrates K=3 knots.

Additional restrictions about the continuity of the polynomialfunctions at each knot can be added, known as the order of the spline,denoted by M. An order M=1 spline indicates that the polynomialfunction fit to each region can be discontinuous at the knots. OrderM=2 restricts piecewise polynomial functions of adjacent regions to becontinuous at their shared knot. M=3 places the additional restrictionthat the functions' first derivative must be continuous at the knots.M=4 places yet another restriction that the functions' second derivativemust be continuous at the knots. We have chosen order M=4 splines,also known as cubic splines, which are widely used (Hastie et al., 2005).The first M columns of T represent the order of the spline (i.e.,continuity restrictions), shown in Eq. (7).

τt = ″i m i, (7)

The subsequent K columns of T represent the polynomial functionapplied to each temperature region, shown in Eq. (8).

τ ξ k Kt = ( ″ − ) , = 1,…,i M k i kM

, + +−1 (8)

One further refinement, used to address erratic behaviour of

polynomials at the extremes where few observations exist, is to placeadditional constraints on the fit of the outer spline regions. Naturalcubic splines restrict the polynomial functions of the outer regions tobe linear beyond the sample boundaries. This added bias at theboundaries is often reasonable considering the sparse number ofobservations. The bottom graph of Fig. 4 illustrates a natural cubicspline fit of aggregate electricity demand to dry-bulb temperature.

Fig. 4. Top: Linear regression line fit to untransformed, outdoor, dry-bulb temperatureobservations. Middle: Fitted regression line for switching regression transformation ofoutdoor, dry-bulb temperature. Temperature break point at 17.9 °C. Bottom: Naturalcubic splines fit of outdoor, dry-bulb temperature fit to aggregate electricity demand.Knots are placed at 3 °C, 23 °C and 30 °C. All three plots are conditioned for visualizationpurposes using a process described by Breheny and Burchett (2015).

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There are three knots placed at 3 °C, 23 °C and 30 °C, selectedempirically using the highest Adjusted R2 as the selection criterion. Asmooth transition between heating and cooling effects is visible around17 °C.

If τ″ is left untransformed during the third step of Algorithm 1, thenthe output of the non-linearity transformation would be a vector ofobservations generated by the first two transformation steps, such that

τT = ″.

4.3.4. Complex temperature transformationsHeating degree-hours (HDH) and cooling degree-hours (CDH) are

derived values which represent the build-up of temperature beyond agiven threshold during a recent window of time. Similar to switchingregression, a temperature break point ξbreak is chosen. HDH isdetermined by summing the number of degrees below ξbreak duringa window of L recent hours, shown in Eq. (9).

∑t τξ i L N= ( − ′ ) , = ,…,i

L

break i,1ℓ=0

−ℓ +(9)

Similarly, CDH is determined by summing the number of degreesabove ξbreak during a window of L recent hours, shown in Eq. (10).

∑t τ ξ i L N= ( ′ − ) , = ,…,i

L

i break,2ℓ=0

−ℓ +(10)

The resulting N L( − ) × 2 transformation matrix T is a piecewiselinear regression, similar to switching regression. Rows i L= 1,…, fromY, X and V must also be removed from the sample. Since CDH andHDH values are approximately linear, we do not fit a model using thesevalues as input to a natural cubic splines transformation.

A finite distributed lag model was initially proposed by Almon(1965) to compute a weighted sum of past explanatory variable effectson a response variable. A more recent implementation of this conceptby Gasparrini et al. (2010); Gasparrini (2011) has come to be known asdistributed lag non-linear models (DLNM). In the DLNM framework,the effects of weather and its relation with time are represented by theconcept of basis. It assumes that the effect at time i is a basis that canbe expressed as a linear combination of exposure and lag transforma-tions of τ′. These transformations are known as basis functions. Forexample, the basis function of temperature's effects may be modelledwith natural cubic splines and is known as the exposure-responseassociation. The weight of the effect may change with time. The basisfunction describing effect weights over time is known as the lag-response association. Together, they comprise the basis known asexposure-lag-response association.

4.4. Metrics for model selection

Having described the space of possible models, we now explain howto choose the best model for the task at hand. For each model, wecompute a measure of variance explained, Adjusted R2. We also checkthe value of the Bayesian Information Criterion (BIC). As Adjusted R2

increases, BIC's value should decrease. If Adjusted R2 decreases andBIC increases or if both Adjusted R2 and BIC increase, then addedexplained variance is not justified by added model complexity (Jameset al., 2013; Ramsey and Schafer, 2012). Additionally, we compute theMean Absolute Error (MAE) and Mean Absolute Percentage Error(MAPE) to indicate out-of-sample predictive power. We seek a modelthat balances explanatory power with out-of-sample predictive power,while being parsimonious and interpretable.

Aside from examining the relationship of each explanatory variablewith aggregate electricity demand individually, the residuals remainingafter fitting a model to data can provide an indication of underlyingissues with the estimated model. The N × 1 vector of residuals shouldbe normally distributed, mean zero and independent of each explana-tory variable.

5. Results for modelling the effects of weather

Table 8 shows the results for each weather model. We also includeseveral trivial models for comparison: a null model (i.e., intercept-only)in which X, V and T have been omitted; non-temperature explanatoryvariables only in which T has been omitted; and dry-bulb temperaturewithout any transformation in which τT = . By comparing all combina-tions of temperature variable transformations and selecting a well-performing model, a substantial amount of variance can be explainedby weather. Though combinations of temperature transformation stepseach produce incremental improvements, the proportion of varianceexplained by any temperature transformations is notable.

Our clearest descriptive results pertain to the time delay betweenobserved temperature and its effects within residential households. Ifan analyst is to use a single temperature observation to explainelectricity demand at time i, the temperature observation at timei − 2 should be used. Of single temperature variables, it also has thehighest Adjusted R2 and out-of-sample predictive power. We interpretthis to mean that residential customer's household thermal controls arereacting to temperatures experienced in the past, not the current hour.

All three temporal transformations which include a window of pastobservations have high Adjusted R2 values and improved out-of-sample prediction accuracy. This suggests that a window of recently-observed temperatures is important to properly describe its relation-ship with electricity demand. Both CDH/HDH and the moving averagetransformations showed that a six-hour window of temperatureobservations yielded the highest Adjusted R2. We feel this validatespart of our hypothesis, that past hours' temperature observations havean effect on the current hour's electricity demand. Notably, despite theprevalence of the CDH/HDH metric in literature, the moving averagetransformation has greater explanatory power and predictive powerin our dataset. This may be caused by the smoothing effect that movingaverage has on the temperature explanatory variable.

The use of heat index and wind chill as components of feels liketemperature has greater Adjusted R2 than the use of dry-bulbtemperature in all cases but two. Our analysis cannot provide addi-tional insight about the underlying process, whether human perceptionor mechanical. Conversely, feels like temperature has less out-of-sample predictive power than dry-bulb temperature. Due to this mixedresult, we reject part of our hypothesis. Namely, the part that statedcoincident weather observations have an effect on electricity demand.We conclude that the feels like temperature transformation has littleadded value over simply using dry-bulb temperature observations.

Despite the strong assumption of linearity made by the switchingregression transformation, it explains untransformed aggregate elec-tricity demand reasonably well using either dry-bulb temperature orfeels like temperature. When estimating unlagged temperature obser-vations, its Adjusted R ≈ 0.852 is comparable to Adjusted R ≈ 0.862

using natural splines. The temperature breakpoint has a straightfor-ward interpretation in relation to electricity demand. The empiricalswitching point for dry-bulb temperature in our data is 17.9 °C.Natural cubic splines do provide more flexibility in modelling thetemperature's non-linear relationship with aggregate electricity de-mand and has higher Adjusted R2 than switching regression. Bothresults support part of our hypothesis, that temperature's effects onelectricity demand are non-linear, having greater impact at low andhigh temperature extremes.

Finally, we comment on the exposure-lag-response transformation,which we have not found in electricity demand analysis literature. Itsintended purpose, to model the weight of an exposure effect over time,is not easily interpretable when applied to our data sample. Combinedwith its minimal improvements to explanatory power and predictionaccuracy, we do not feel its use is justified.

Based on these results, we select the model that uses dry-bulbtemperature, combined with the six-hour moving average and thenatural spline transform. This model obtained an Adjusted R2 of 0.902.

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A similar model that uses feels-like temperature instead has a slightlyhigher Adjusted R2 of 0.904 but lower out-of-sample predictive power.Furthermore, a similar model with exposure-lang-response transfor-mation rather than the six-hour moving average has an even higherAdjusted R2 of 0.910, but as mentioned above, is not easily inter-pretable.

Finally, we comment on the residuals of the selected model, whichfulfill the first assumption of linear regression analysis, that errors benormally distributed with mean zero. The next step of residual analysisis to assess heteroscedasticity of residuals. The plots in Fig. 5 indicatethat heteroscedastic and autocorrelation consistent (HAC) standarderrors must be used when performing hypothesis tests in the TOUpricing case study (Zeileis, 2004). The first plot shows increasedvariability of residuals at warm temperatures. The middle plot showsgreater variance associated with summer and winter seasons. This islikely a result of temperature and season's collinearity with dry-bulbtemperature. This result is supported quantitatively by the Durbin-

Watson statistic in Table 8. A Durbin-Watson value <1 indicatespositive serial correlation of residuals Bhargava et al. (1982). Thebottom plot illustrates that variance of residuals increases with largervalues of the response variable. This too is likely related to tempera-ture. Because warmer temperatures are associated with higher elec-tricity demand, it follows that greater residual variance is associatedwith higher electricity demand.

6. Methodology for TOU impact analysis

In Section 4, we described the methodology for modelling time,price and weather explanatory variables as the matricesX, V and T. Weshow Eq. (2) below again for clarity, since the same form will be usedduring the TOU impact analysis.

β ω θβY X V T= + + +0

We set the temperature transformation matrix T to be a dry-bulb,

Table 8Results of temperature transformation comparison. The first three columns show how temperature transformations from the three categories are combined. Adjusted R2 column is ourprimary evaluation criterion. BIC and Durbin-Watson columns provide secondary measures of model complexity and serially correlated errors. The final two columns report predictiveaccuracy using average MAE and average MAPE measured using time series cross-validation (Hyndman and Fan, 2010). The model identified in Section 5 as having the greatestexplanatory power, out-of-sample prediction accuracy, and interpretability has been bolded.

Temporal transform Weathertransform

Non-linearity transform Adj. R2 BIC Durbin-Watson

Avg. MAE(kWh)

Avg. MAPE (%)

ideal=1 ideal=low ideal=2 ideal=0 ideal=0, max=100Null Model (i.e., intercept only) 0.000 22339.7 0.056 0.432 37.57Non-Temperature Explanatory Variables Only 0.438 14497.9 0.042 0.397 37.36None (i−0) None (Drybulb) None (Linear) 0.580 10324.2 0.060 0.266 23.51None (i−0) None (Drybulb) Switching Regression 0.854 −4822.3 0.201 0.166 14.20None (i−0) None (Drybulb) Natural Splines 0.862 −5551.9 0.198 0.157 13.19None (i−0) Feels Like Switching Regression 0.857 −5097.0 0.208 0.164 14.00None (i−0) Feels Like Natural Splines 0.862 −5550.1 0.210 0.158 13.35i−1 None (Drybulb) Switching Regression 0.875 −7068.5 0.229 0.149 13.05i−1 None (Drybulb) Natural Splines 0.884 −8028.3 0.228 0.141 12.02i−1 Feels Like Switching Regression 0.878 −7326.9 0.236 0.149 12.82i−1 Feels Like Natural Splines 0.884 −8036.4 0.241 0.143 12.19i−2 None (Drybulb) Switching Regression 0.881 −7741.3 0.258 0.144 12.71i−2 None (Drybulb) Natural Splines 0.889 −8722.6 0.262 0.135 11.64i−2 Feels Like Switching Regression 0.883 −7974.6 0.266 0.144 12.46i−2 Feels Like Natural Splines 0.890 −8810.8 0.276 0.137 11.76i−3 None (Drybulb) Switching Regression 0.873 −6803.5 0.248 0.149 12.96i−3 None (Drybulb) Natural Splines 0.880 −7627.2 0.250 0.138 11.72i−3 Feels Like Switching Regression 0.875 −7026.7 0.257 0.149 12.73i−3 Feels Like Natural Splines 0.882 −7807.8 0.266 0.139 11.81i−4 None (Drybulb) Switching Regression 0.853 −4683.5 0.232 0.159 13.50i−4 None (Drybulb) Natural Splines 0.859 −5294.1 0.231 0.147 12.05i−4 Feels Like Switching Regression 0.855 −4925.3 0.240 0.158 13.28i−4 Feels Like Natural Splines 0.862 −5563.3 0.244 0.147 12.13i−5 None (Drybulb) Switching Regression 0.825 −2171.8 0.192 0.173 14.29i−5 None (Drybulb) Natural Splines 0.830 −2628.2 0.191 0.160 12.84i−5 Feels Like Switching Regression 0.828 −2437.6 0.199 0.172 14.11i−5 Feels Like Natural Splines 0.834 −2940.9 0.201 0.161 12.94i−6 None (Drybulb) Switching Regression 0.790 387.4 0.166 0.189 15.39i−6 None (Drybulb) Natural Splines 0.796 −9.7 0.165 0.179 14.34i−6 Feels Like Switching Regression 0.794 114.4 0.171 0.188 15.24i−6 Feels Like Natural Splines 0.801 −321.1 0.172 0.179 14.41CDH/HDH (L=6) None (Drybulb) Switching Regression 0.895 −9493.4 0.183 0.133 11.53CDH/HDH (L=6) None (Drybulb) Natural Splines N/A N/A N/A N/A N/ACDH/HDH (L=6) Feels Like Switching Regression 0.896 −9629.3 0.184 0.134 11.36CDH/HDH (L=6) Feels Like Natural Splines N/A N/A N/A N/A N/AMoving Avg. (L=6) None (Drybulb) Switching Regression 0.895 −9492.1 0.195 0.139 12.33Moving Avg. (L=6) None (Drybulb) Natural Splines 0.902 −10537.5 0.196 0.128 10.94Moving Avg. (L=6) Feels Like Switching Regression 0.897 −9771.0 0.193 0.138 11.99Moving Avg. (L=6) Feels Like Natural Splines 0.904 −10718.9 0.199 0.130 11.16Lag-Response: Cubic

Polynomial (L=6)None (Drybulb) Exposure-Response: Switching

Regression0.902 −10388.5 0.197 0.127 10.93

Lag-Response: CubicPolynomial (L=6)

None (Drybulb) Exposure-Response: NaturalSplines

0.910 −11559.9 0.209 0.118 9.86

Lag-Response: CubicPolynomial (L=6)

Feels Like Exposure-Response: SwitchingRegression

0.901 −10257.2 0.192 0.129 10.88

Lag-Response: CubicPolynomial (L=6)

Feels Like Exposure-Response: NaturalSplines

0.911 −11732.2 0.213 0.123 10.43

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six-hour moving average, natural cubic splines transformation.Because collinearity of temperature and seasonal explanatory variablesis not a concern when analyzing the effects of TOU, we add acategorical explanatory variable for month to X, supported by

ANOVA, such that the categorical variables in X are:

x p•, = 1 through x p•, = 11 are month indicators representing Januarythrough December.

x p•, = 12 through x p•, = 34 are hour-of-day indicators representing 00-:00 through 23:00.

x p•, = 35 is a working day indicator.x p•, = 36 through x p•, = 58 are indicators representing the hour-of-da-

y×working day interaction.

To model effects associated with TOU pricing with hourly fidelity,the time and temperature matrices X and T are then held constant. Wewill use backward selection, ANOVA and HAC standard errors toremove insignificant variables from a saturated matrix V of explanatoryvariables related to price. Backward selection starts with all possibleexplanatory variables in X, V and T. All two-way and three-wayinteractions combining a TOU billing indicator, working day, hour-of-day and utility rate season are included. This initial model is alsocalled the saturated model. The variable interactions provide thenecessary degrees of freedom to explain the effects of TOU billing foreach hour of day. We remove variables with the largest p-value (i.e., theleast statistically significant variable) one at a time until the analysis ofvariance stopping condition is met (James et al., 2013). The remaining,significant explanatory variables in V are used as components of themultiple regression model in a “what if” analysis. We use the resultsfrom the “what if” analysis to quantify the change in demand associatedwith TOU pricing. The categorical variables in V before backwardselection are:

v p•, = 1 is a utility rate season indicator representing summer andwinter rates.

v p•, = 2 is a TOU active indicator representing whether customers arebilled according to flat rates or TOU rates.

v p•, = 3 through v p•, = 25 are indicators representing the hour-of-da-y×rate season interaction.

v p•, = 26 through v p•, = 48 are indicators representing the hour-of-da-y×TOU active interaction.

v p•, = 49 is an indicator representing the working day×rate season i-nteraction.

v p•, = 50 is an indicator representing the working day×TOU active i-nteraction.

v p•, = 51 is an indicator representing the rate season×TOU active in-teraction.

v p•, = 52 through v p•, = 74 are indicators representing the hour-of-da-y×working day×rate season interaction.

v p•, = 75 through v p•, = 97 are indicators representing the hour-of-da-y×working day×TOU active interaction.

v p•, = 98 through v p•, = 120 are indicators representing the hour-of-da-y×rate season×TOU active interaction.

v p•, = 121 is an indicator representing the working day×rate season×-TOU active interaction.

Using backward selection, we remove variables from the saturatedmatrix V to create a more parsimonious model. The following inter-actions cannot be justified by ANOVA and are dropped from V:working day×rate season×TOU active, hour-of-day×workingday×TOU active and working day×TOU active. The model used in thiscase study yields Adjusted R = 0.9352 .

Algorithm 3. “What if” analysis used to quantify the effects ofmandatory TOU electricity pricing.

1. Fit a model to the entire sample of data. Our sample runs fromMarch 1, 2011 – October 17, 2012.

Fig. 5. Top: Residuals as a function of dry-bulb temperature observations, resultingfrom a comparison model using dry-bulb, six-hour moving average and natural cubicsplines to generate the temperature transformation matrix T. Middle: Residuals as afunction of time, resulting from a comparison model using dry-bulb, six-hour movingaverage and natural cubic splines to generate the temperature transformation matrix T.Bottom: Residuals as a function estimated response, resulting from a comparison modelusing dry-bulb, six-hour moving average and natural cubic splines to generate thetemperature transformation matrix T.

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2. For the summer utility rate season, select subset where TOUactive=FALSE (i.e., May 2011 – October 2011).

3. Group the selected observations by working day indicator. For eachworking day type find the mean electricity demand observations foreach hour. These hourly averages for each working day typerepresent the observed summer.

4. Copy the selected sample data from step 2 into a new hypotheticalsample of data called the counterfactual summer.

5. In the counterfactual summer, change the TOU active indicator fromFALSE to TRUE.

6. Estimate a response vector using the adjusted counterfactualsummer from step 5. Because TOU active has been changed toTRUE, the coefficients estimated in step 1 will create a responsevector as if TOU billing had been active during summer 2011. Thisestimated response vector is the “what if” analysis.Algorithm 3 describes our “what if” methodology, similar to that

used by Navigant Research and Ontario Energy Board (2013).However, because our sample of data does not have a complete winterutility rate season of TOU active=FALSE from November 2010 throughMay 2011, we are only able to carry out the analysis for summer utilityrate season.

7. Results for TOU impact analysis

The average demand change during summer on-peak and mid-peakperiods is −2.6% and −2.4% respectively. This translates to−0.035 kWh ( ± 0.024 kWh) per household each hour during on-peakperiods and −0.030 kWh ( ± 0.024 kWh) change during mid-peakperiods. Changes during working day and non-working day off-peak

periods are −0.9% and −0.6% but are not statistically significant.Table 9 summarizes the hourly effects averaged by TOU price period.Fig. 6 shows this same information graphically. The estimated effects ofTOU pricing for each hour are plotted over coloured regions represent-ing the three price periods of a summer working day.

The results from Table 9 can be extrapolated to all 20,556residential customers in the local distribution company's serviceregion. Demand during each on-peak hour would change by −0.72MWh ( ± 0.49 MWh), mid-peak hours would change by −0.62 MWh (± 0.49 MWh), off-peak would change by −0.23 MWh ( ± 0.49 MWh),and each hour of non-working days would change by −0.17 MWh ( ±0.62 MWh).

We study the daily peak-to-average ratio since it is a metric oftenused by utilities to measure how extreme demand fluctuations are.Each day's peak-to-average ratio is defined as the peak demand for theday divided by the average demand during that day. The averageobserved peak-to-average ratio for summer 2011 under flat pricing was1.441. The estimated summer peak-to-average ratio of the counter-factual sample is 1.429. This represents an estimated change of−0.844% to the peak-to-average ratio, with a 95% confidence intervalof ± 0.6%.

The local distribution company's peak hour observed during thepre-TOU summer occurred on Thursday, July 21, 2011 at 18:00 EDT,averaging 3.54 kWh per household. Using estimated demand from the“what if” analysis, on-peak TOU pricing would have reduced theaverage household consumption during that hour to 3.42 kWh ( ±0.03 kWh), a reduction of 3.4%. The most extreme peak-to-averageratio was observed on Tuesday, June 21, 2011 with a value of 1.65. HadTOU pricing been in place that summer, the estimated peak-to-averageratio on that date would have been 1.57, a reduction of 4.8%.

8. Conclusions and policy implications

The main policy implication of this paper is the introduction of amethodology that energy researchers and practitioners may use tomodel residential demand, including the effects of weather, whenanalyzing the impact of pricing strategies such as TOU. Our methodol-ogy evaluates a wide variety of approaches used to cope with the effectsof time, weather and price, and selects the best model based onexplanatory power, out-of-sample prediction accuracy, interpretabilityand goodness of fit. These effects can vary greatly by region, so no

Table 9Estimated change in average household electricity demand for each TOU price period.

Summerprice period

Hourlyimpact(kWh)

95% Conf.interval(kWh)

Hourlyimpact (%)

95% Conf.interval (%)

On-Peak −0.035 ± 0.024 −2.641 ± 1.819Mid-Peak −0.030 ± 0.024 −2.403 ± 1.933Off-Peak −0.011 ± 0.024 −0.888 ± 1.901Non-Working

Day−0.009 ± 0.030 −0.617 ± 2.212

Fig. 6. The hourly effects of a “what if” analysis estimated using summer 2011 data from our sample. The observed data is the solid, black line, indicating the mean of observed demandfor each hour of working days. The dotted blue line indicates the mean of estimated demand for each hour of working days, had TOU billing been in place. A 95% confidence interval isalso plotted for each hour.

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single residential electricity demand model is universally applicable.For this reason, policy makers should insist on clear, reproduciblemethodologies such as ours, which include details about variableselection and model assessment measures. The results of an analysisshould only be considered reliable if adequate supporting metrics areprovided.

Furthermore, policy makers should strive to make more electricitydemand data available in order to validate new pricing schemes andconservation programs; our analysis would not have been possiblewithout access to a large smart meter dataset.

The second policy implication stems from our TOU impact analysis.We conclude that TOU helped mitigate peak electricity demand,reducing summer on-peak demand by 2.6%. Our findings are consis-tent with those of Newsham and Bowker (2010), which estimates thatTOU implementations typically see on-peak demand reductions <5%.However, we observe that the estimated effects in our dataset are lesspronounced than initial results elsewhere in Ontario. Faruqui et al.(2013b) estimated first-year results from four Ontario TOU programswith summer on-peak reductions in the range 2.6–5.7%. Our resultfalls at the bottom of that range. Our result is also lower than that ofNavigant Research and Ontario Energy Board (2013), which analyzed asample of 10,000 residential consumers in various locations withinOntario, finding a summer on-peak reduction of 3.3%. It is worthnoting that Navigant's 3.3% demand reduction estimate falls within our95% confidence interval.

Both the slight decrease in peak-to-average ratio and the hourlydemand reduction across all TOU price periods indicate that manda-tory TOU pricing can achieve electricity conservation. However,analysis of electricity demand shifting is more complex. Table 9 showsthat the majority of estimated summer demand reduction occurs in on-peak and mid-peak periods. Change during off-peak periods for bothworking days and non-working days is minimal. We interpret this tomean that electricity demand is not being shifted to off-peak periods,but is only being conserved. Conservation is focused during on- andmid-peak periods. Given this finding, the local distribution companyshould adjust its long term forecasts. If conservation is the trend inmany other local distribution companies, the province might be able todefer construction of new generation facilities. If demand shifting weremore substantial (e.g., an increase during off-peak periods) it wouldresult in a flattened demand curve. If such a trend were to exist in manylocal distribution companies, then the make-up of generation facilitiesthroughout the province could shift from those with fast ramp rates,such as natural gas or reservoir hydroelectric, to those that are moreconstant, such as nuclear.

The demand reduction during the off-peak hours of 19:00 through21:00 during working days is counter-intuitive. When the hours 17:00through 18:00 during the second mid-peak period are also considered,demand reduction seems focused during the evening after typical workhours. Residential customers may be attempting to conserve electricity,but they may only have flexibility in their after-work household activity.Because Ontario's TOU pricing also applies to commercial customers, itmay not be optimally structured around residential demand flexibility.This misalignment was also noted by Adepetu et al. (2013) in theirresults when studying aggregate provincial data. The province ofOntario could study the impact of placing commercial and residentialcustomers on separate TOU schedules and adjusting rates accordingly.If Ontario's residential TOU rate schedule remains unchanged, there isan opportunity for technology companies in the realm of connecteddevices. This result suggests that residential customers, unaided byautomated devices, have difficulty reacting to TOU rates when outsidethe home. Devices and software which can incorporate the user's TOUrate schedule could reduce the household electricity bill and associatedon-peak emissions to a greater extent.

Because our sample of data is from one local distribution companyin south west Ontario, we acknowledge that our results are only directlyapplicable to that region. Additionally, because we only have data for

one summer of before and after the switch to TOU pricing, we cannotassess the effects of TOU pricing during winter rates. We restate ouroriginal question in this context: Is Ontario's mandatory TOU policyassociated with energy conservation or load shifting during the winterrate season in this local distribution company's service region?

Acknowledgements

This research was funded by the Natural Sciences and EngineeringResearch Council of Canada (Rosenberg) and the Waterloo Institute forSustainable Energy/Cisco Smart Grid Research Fund (Golab).

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